4E : The Quantum Universe. Lecture 6, April 6 Vivek Sharma

Size: px

Start display at page:

Download "4E : The Quantum Universe. Lecture 6, April 6 Vivek Sharma"

Aubrie O’Connor’
6 years ago
Views:

1 4E : The Quantum Universe Lecture 6, April 6 Vivek Sharma modphys@hepmail.ucsd.edu

2 Bragg Scattering photographic film

3 Bragg Scattering: Probing Atoms With X-Rays Incident X-ray detector Constructive Interference when net phase difference is 0, π etc This implied path difference traveled by two waves must be integral multiple of wavelength : nλ=dsinϑ From X Ray (EM Wave) Scattering data, size of atoms was known to be about m

4 X-Ray Picture of a DNA Crystal and Discovery of DNA Structure!

5 Where are the electrons inside the atom? Early Thought: Plum pudding model Atom has a homogenous distribution of Positive charge with electrons embedded in them e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - e - Positively charged matter or +? How to test these hypotheses? Shoot bullets at the atom and watch their trajectory. What Kind of bullets? + Core Indestructible charged bullets Ionized He ++ atom = α ++ particles Q = +e, Mass M α =4amu >> m e, V α = x 10 7 m/s (non-relavistic) [charged to probe charge & mass distribution inside atom]

6 Plum Pudding Model of Atom Non-relativistic mechanics (Vα/c = 0.1) In Plum-pudding model, α-rays hardly scatter because Positive charge distributed over size of atom (10-10m) Mα >> Me (like moving truck hits a bicycle) Æ predict α-rays will pass thru array of atoms with little scatter (~1o) Need to test this hypothesis Æ Ernest Rutherford 6

7 Rutherford Scattering discovered by his PhD Student (Marsden)

8 Force on α-particle due to heavy Nucleus Outside radius r =R, F Q/r Inside radius r < R, F q/r = Qr/R Maximum force at radius r = R α particle trajectory is hyperbolic Scattering angle is related to impact par. kq Q θ = α Impact Parameter b cot mv α α

9 Rutherford Scattering: Prediction and Experimental Result n = 4 kzenna 1 4 α α ϕ 4 R m v Sin ( /) # scattered Vs φ depends on : n = # of incident alpha particles N = # of nuclei/area of foil Ze = Nuclear charge K α of incident alpha beam A= detector area

10 Rutherford Scattering & Size of Nucleus distance of closest appoach r size of nucleus 1 Kinetic energy of α = K = mv α α β nucleus nucleus α particle will penetrate thru a radius r until all its kinetic energy is used up to do work AGAINST the Coulomb potential of the Nucleus: 1 mv MeV= k K α = α β = 8 r = kze K For K =7.7.MeV, Z = 13 α α Al kze r = = K α Size of Nucleus = 10 Size of Atom = m ( Ze)( e) m r 15 m

11 Dimension Matters! Size of Nucleus = 10 Size of Atom = m m How are the electrons located inside an atom? How are they held in a stable fashion? necessary condition for us to exist! All these discoveries will require new experiments and observations

12 Where are the Electrons in an Atom?

13 Clues: Continuous & Discrete spectra of Elements

14 Visible Spectrum of Sun Through a Prism

15 Emission & Absorption Line Spectra of Elements

16 Kirchhoff Experiment : D Lines in Na D lines darken noticeably when Sodium vapor introduced Between slit and prism

17 Emission & Absorption Line Spectrum of Elements Emission line appear dark because of photographic exposure Absorption spectrum of Na While light passed thru Na vapor is absorbed at specific λ

18 Spectral Observations : series of lines with a pattern Empirical observation (by trial & error) All these series can be summarized in a simple formula = R, n, 1,,3,4.. f > ni ni = λ nf n i Fitting to spectral line series data R= m 7 1 How does one explain this?

19 The Rapidly Vanishing Atom: A Classical Disaster! Not too hard to draw analogy with dynamics under another Central Force Think of the Gravitational Force between two objects and their circular orbits. Perhaps the electron rotates around the Nucleus and is bound by their electrical charge MM 1 F= G k r QQ r 1 Laws of E&M destroy this equivalent picture : Why?

20 Bohr s Bold Model of Atom: Semi Quantum/Classical F +e Ur () KE = r e = k r 1 m v e -e m e V 1. Electron in circular orbit around proton with vel=v. Only stationary orbits allowed. Electron does not radiate when in these stable (stationary) orbits 3. Orbits quantized: M e v r = n h/π (n=1,,3 ) 4. Radiation emitted when electron jumps from a stable orbit of higher energy stable orbit of lower energy E f -E i = hf =hc/λ 5. Energy change quantized f = frequency of radiation

21 General two body motion under a central force Reduced Mass of -body system -e m e +e F V reduces to r m e Both Nucleus & e - revolve around their common center of mass (CM) Such a system is equivalent to single particle of reduced mass µ that revolves around position of Nucleus at a distance of (e - -N) separation µ= (m e M)/(m e +M), when M>>m, µ=m (Hydrogen atom) Νot so when calculating Muon (m µ = 07 m e ) or equal mass charges rotating around each other (similar to what you saw in gravitation)

22 Allowed Energy Levels & Orbit Radii in Bohr Model 1 e E=KE+U = mv e k r Force Equality for Stable Orbit Coulomb attraction = CP Force Total Energy mv e e = k r Negative E Bound system This KE = e k r mv e r E = KE+U= - k e r much energy must be added to the system to break up the bound atom = Radius of Electron Orbit : mvr = n n v =, mr 1 ke substitute in KE= mv e = r n rn =, n = 1,,... mke n = 1 Bohr Radius a 1 10 a0 = = m mke In ge r = na n = neral n 0; 1,,... Quantized orbits of rotation 0

23 E = K + U = n since n Energy Level Diagram and Atomic Transitions 0 Interstate transition: n 0 f i ke r,n =quantum number ke 13.6 En = = ev, n = 1,,3.. an n E ke 1 1 = a 0 ni n f r = a n = hf = E f ke 1 1 = ha 0 n n f ke = = λ c hca 0 n = R i E f 1 1 n n i n f i f i n f

24 Hydrogen Spectrum: as explained by Bohr E n = ke a 0 Z n Bohr s R same as Rydberg Constant R derived emperically from spectral series

25 A Look Back at the Spectral Lines With Bohr s Optic E n = ke a 0 Z n Rydberg Constant

26 Bohr s Atom: Emission & Absorption Spectra photon photon

27 Some Notes About Bohr Like Atoms Ground state of Hydrogen atom (n=1) E 0 = ev Method for calculating energy levels etc applies to all Hydrogenlike atoms -1e around +Ze Examples : He +, Li ++ Energy levels would be different if replace electron with Muons Bohr s method can be applied in general to all systems under a central force (e.g. gravitational instead of Coulombic) QQ M M If change U( r) = k G r r Changes every thing: E, r, f etc 1 1 "Importance of constants in your life"

Physics 2D Lecture Slides Lecture 15: Feb 2nd Vivek Sharma UCSD Physics. Where are the electrons inside the atom?

Physics 2D Lecture Slides Lecture 15: Feb 2nd Vivek Sharma UCSD Physics. Where are the electrons inside the atom? Physics D Lecture Slides Lecture 15: Feb nd 005 Vivek Sharma UCSD Physics Where are the electrons inside the atom? Early Thought: Plum pudding model Atom has a homogenous distribution of Positive charge